I am now convinced by Shipman, Machover,
Feferman that Objectivity is the fundamental
issue that arises in an evaluation of Lakatos'
thesis that conclusive proofs are unlikely.
At the same time I am wondering about the
level of awareness of the very great extent
political agenda and ideology influence and
disturb Lakatos' thinking
(as it did Popper's)? I am wondering to
what extent Sol Feferman is aware of the
political content of his conviction that
mathematics is a social construct? [How
he sides with (early) Vico against Descartes?]
[Steve Simpson tried to get STAN ROSEN
on board FOM; I am venturing these remarks,
and cc'ing them to Stan, in the hope that
he might be inspired to participate.]
Certainly Reuben Hersh's new book
WHAT IS MATHEMATICS, REALLY? makes clear
the presence of political agendas. (I
myself am politically naive, for example,
it came as a terrible revelation to me when
Noam Chomsky pointed out the powerful
anti-Enlightenment motives driving Quine's
philosophy.)
BTW the discussion on another thread of
FOM, on general intellectual interest is of
considerable moment in the light of the
nuclear wars raging in liberal arts colleges
over the place of math and science in them.
The issues go back to the birth of the
Humanities with Petrarch, and arise
horrifically in the mid 19th century when
the Scot philosopher William Hamilton sought
to purge the universities of mathematics on
the grounds (to put it one way) that mathematics
was of not general (humanitarian) interest. I
can speak from bitter experience that the
Petrarchian-Hamiltonian tradition is very
much alive in some top 30 liberal arts colleges.
I am uncertain of the extent to which FOMers
are interested in these political agends. E.g.
does Steve Simpson's following of Ayn Rand affect
his thinking about foundations and philosophy
of mathematics? I will confine myself to
a couple observations connected with Lakatos.
Perhaps someone more knowing -- like Stan Rosen --
would chime in.
[1] Plato privileged mathematical knowledge, True
Platonism does not concern existence, but the
definitiveness and exemplary character of
mathematical knowledge. Only those who could
attain mathematical insight and understanding
could become wise leaders.
The very thought that there can be privileged
insight not generally attainable, and this the
grounds for authority, was anathema to, e.g.,
Popper, as to Lakatos. (See Hersh's distinction
between aristocratic and humanitarian mathematics.)
Lakatos is out to humble mathematics by eroding
what (to some) is distinctive about it.
N.B., it was quite within Lakato's power
to explicitly recognize in the opening pages
of Prfs&Ref. that one can have a conclusive
and Cauchy style proof of Euler's formula for
spheroid polyhedra; see for example Hilbert's
very well known treatment in Geometry and the
Imagination, for simple [spheroid] polyhedra.
Lakatos is not up front about this. When on
p.30 the issue does arise, and Beta says
"not guesswork, but insight", Teacher
in a towering rage exclaims, "I abhor your
pretentious 'insight'. I respect conscious
guessing, because it comes from the best
human qualities: courage and modesty."
[2] it is worth remarking that the story of
Brouwer's attempt to wrest the leadership of
mathematics away from Hilbert points to a
profoundly problematic conception of science
we hopefully have gotten away from. That a
science needs a great leader, one who has
high and priv. insight. Note the opening
of Hilbert's Axiomatic Thinking. The
writings of Heidegger circa 1929 are
horrifyingly dominated by the leadership
concept. For example, he remarks that
biology will never be a fundamental science
until a great leader arises (or is allowed
to arise). And of course the last third of
Being and Time is devoted to creating space
in which great leader may arise.
[3] In Indiscrete Thoughts, Gian-Carlo
Rota associates mathematical beauty with
insightfulness. That is very much
against the temper of Lakatos and the
spirit of modern mathematics. But
I think that there is such insightfulness
that there is "objective" conclusiveness
in mathematics; and that this need
not have any political consequences
whatsoever.
Let's take the philosophy of
mathematics out of the smoke-filled
back rooms, or fern bars.
rbrt tragesser